According to the extreme ray characterization of diagonally dominant matrices by Barker and Carlson [1]}, a symmetric matrix \(M \in DD_n\)  if and only if it can be written as \( M = \sum _{i=1}^{n^2} \alpha _i v_iv_i^T, \; \alpha _i \in \mathbb {S}^1\) , where \( \lbrace  v_i\rbrace \)  is the set of all nonzero vectors in \(\mathbb {R}^n\)  with at most 2 nonzero components, each equal to \( \pm 1\) . The vectors \( \lbrace  v_i\rbrace \)  are the extreme rays of the \( DD_n \)  cone. A similar decomposition is available for scaled diagonally dominant matrices. According to [2]}, we know that any scaled diagonally dominant matrix \(M\)  can be written as \( M = \sum _{i=1}^{\binom{n}{2}} V_i \Lambda _i V_i^T \) , where \(V_i\)  is an \(n \times 2 \)  matrix whose columns each contain one nonzero element which is equal to 1, and \(\Lambda _i \in \mathbb {S}^2_+\) .
